How to find the solution for a system of equations - PSAT Math

The most simple method for solving systems of equations is to transform one of the equations so it allows for the canceling out of a variable. In this case, we can multiply by to get .

Then, we can add to this equation to yield , so .

We can plug that value into either of the original equations; for example, .

So, as well.

In the second equation, you can substitute for from the first.

Now, substitute 2 for in the first equation:

The solution is

By solving one equation for , and replacing in the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

Add the equations together.

Put the terms together to see that .

Substitute this value into one of the original equaitons and solve for .

Now we know that , thus we can find the sum of and .

To find the point where these two lines intersect, set the equations equal to each other, such that is substituted with the side of the second equation. Solving this new equation for will give the -coordinate of the point of intersection.

Subtract from both sides.

Divide both sides by 2.

Now substitute into either equation to find the -coordinate of the point of intersection.

With both coordinates, we know the point of intersection is . One can plug in for and for in both equations to verify that this is correct.

Solve the second equation for , allowing us to solve using the substitution method.

Substitute for in the first equation, and solve for .

Now, substitute for in either equation; we will choose the second. This allows us to solve for .

Now we can write the solution in the notation , or .

The number of pounds of coffee beans totals 50, so one of the equations would be

.

The total price of the Kona beans, is its unit price, $24 per pound, multiplied by its quantity, pounds. This is dollars. Similarly, the total price of the Ethiopian delight beans is dollars, and the price of the mixture is dollars. Add the prices of the Kona and Ethiopian Delight beans to get the price of the mixture:

We are trying to solve for in the system

Multiply the second equation by , then add to the first:

The mixture includes 15 pounds of Kona beans.

To solve this problem, you must first solve the system of equations for both and , then plug the values of and into the final equation.

In order to solve a system of equations, you must add the equations in a way that gets rid of one of the variables so you can solve for one variable, then for the other. One example of how to do so is as follows:

Take the equations. Multiply the first equation by two so that there is (this will cancel out the in the second equation).

Add the equations:

Find the sum (notice that the variable has disappeared entirely):

Solve for .

Plug this value of back into one of the original equations to solve for :

Now, plug the values of and into the final expression:

The answer is .

Reduce the second system by dividing by 3.

Second Equation:

We this by 3.

Then we subtract the first equation from our new equation.

First Equation:

First Equation - Second Equation:

Left Hand Side:

Right Hand Side:

Our result is:

We add the two systems of equations:

For the Left Hand Side:

For the Right Hand Side:

So our resulting equation is:

Divide both sides by 10:

For the Left Hand Side:

For the Right Hand Side:

Our result is:

We first multiply the second equation by 4.

So our resulting equation is:

Then we subtract the first equation from the second new equation.

Left Hand Side:

Right Hand Side:

Resulting Equation:

We divide both sides by -15

Left Hand Side:

Right Hand Side:

Our result is:

We first add both systems of equations.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

We divide both sides by 3.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

If we multiply both sides of our bottom equation by , we get . We can now add our two equations, and eliminate , leaving only one variable. When we add the equations, we get . Therefore, . Finally, we go back to either of our equations, and plug in so we can solve for .